We investigate the bipartite and tripartite entanglement and the mechanical ground-state cooling in a hybrid microwave-optomechanical circuit. This system, which consists of a common microwave $LC$ resonator, two single-Cooper-pair transistors (SCPTs), and two micromechanical resonators, when driven in a specific regime, behaves equivalently as an optomechanical system with a single cavity field interacting with two mechanical resonators. The cavity couples to the mechanical modes via radiation-pressure, cross-Kerr (CK) type nonlinearities, and an interaction between the two mechanical resonators. First, we demonstrate that adjusting the gate-charge deviation and the detuning frequency enables manipulation of multipartite entanglement within this system. Then, our results indicate that distinct types of CK couplings enhance either the cooling efficiency or the entanglement generation. More specifically, the three-mode CK coupling contributes to multipartite entanglement, whereas the conventional CK coupling plays a key role in simultaneous ground-state cooling of the mechanical modes. This observed competition between cooling efficacy and entanglement formation yields insights for the optimal design of hybrid quantum systems built for multitask operations.

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• Light-matter interaction
• Optomechanics
• Quantum optics